Dirichlet ' s Theorem

$$p / ( p-1 ) ! + \left(\frac{a}{p}\right)  a^{(p-1)/2}$$

Proof.
Consider the equation $Ax \equiv a \bmod p$ with $A, x \in \{ 1,2, \cdots p-1\}$,

Case $\left(\frac{a}{p}\right)=-1$
In this case $x$ and $A$ are different members of the set $\{ 1,2, \cdots p-1\}$, there are $(p-1)/2$ distinct pairs $(A, x) $ and pairwise multiplication gives the following identity: $( p-1 ) ! = a^{(p-1)/2}$.

Case $\left(\frac{a}{p}\right)=1$
In this case $a$ is a quadratic residue of $p$ so there are two pairs where $x$ and $A$ are equal members of the set $\{ 1,2, \cdots p-1\}$, there are $(p-3)/2$ distinct pairs $(A, x) $ and pairwise multiplication gives the following identity: $\frac {( p-1 ) ! }{k (p-k)}= a^{(p-3)/2}$.
Now $k( p-k) = kp - k^2 \equiv -a \bmod p $.  Another pairwise multiplication gives the following identity: $( p-1 ) ! = - a^{(p-1)/2}$.

Combining both cases and replacing the sign with the Legendre symbol gives $$p / ( p-1 ) ! + \left(\frac{a}{p}\right)  a^{(p-1)/2}.$$

Epiphany

I had one again. I am still in awe. I used to call them "cognitions", but that's not enough. The right word is epiphany. The protocol for reading maths is to decipher what the author is trying to say. Often mathematical ideas are simple to visualize but notoriously hard to write on paper. When you have deciphered the text the visualisation is implanted and suddenly it all makes sense. And when the idea is particularly beautiful the implant is experienced as an epiphany.
( Epiphany : An illuminating realization or discovery, often resulting in a personal feeling of elation, awe, or wonder. )

Exercise

Let $ p \in \mathbb {Z}[X] $ and of fifth degree with only the terms for $x^5$, $x^4$ and $x^3$ known, the terms for $x^2$, $x$ and $1$ are not known. $$ p=x^5 - 5x^4 - 35x^3 + \cdots $$ The ( five ) roots of $p $ form an arithmetic sequence. Find the roots of $p $.

Disquisitiones Arithmeticae

It's time to read Gauss's original work. More later.

A Five By Five Magic Square

A few years ago I wanted to find a 5 by five magic square. I gave up on it. See this post for the results I came up with in 2011. I used the wrong tools, I can see that clearly now.

Yesterday and today I worked on it. Again without finding a solution. This morning I thought "Should I spend another day on it? What am I doing with my time? I am not going to solve this one ( either )". But problems you can't solve become little traumas in your subconscious. From time to time they remind you that you weren't able to crack them. It hurts. Probably more than you can imagine.

There was one thing I could add to my search algorithm, I decided to add that and then stop. I had only one diagonal left that didn't add up to 65. Anyway, the last thing worked and I found one. One hurting trauma less.

$$
\begin{array}{ccccc}
2 & 25 & 5 & 20 & 13 \\
18 & 4 & 17 & 7 & 19 \\
8 & 24 & 22 & 10 & 1 \\
23 & 9 & 6 & 16 & 11 \\
14 & 3 & 15 & 12 & 21
\end{array}
$$

The integers $1,2 \dots 25$ are laid out in a 5 by 5 matrix such that the rows, columns and the two diagonals total to 65.

Now the real fun begins: finding bigger ones, special ones maybe, and analyzing and optimizing my method. Where are the limits? And questions, many questions like how many different 5 by 5s are there? Is there one with the number 13 in the centre?

Explained: Mathematical Virus type MV/C

ARE YOU INFECTED WITH MV/C ?

What is MV

A mathematical virus (MV) is a preconception about the structure, function or method of mathematics which impairs one's ability to do mathematics.

What is MV/C?

MV/C is a mathematical virus which is easily diagnosed, the infected suffer from the delusion that "Coordinates are essential to calculations". Physicists and engineers are especially susceptible to this virus, because most of their textbooks are infected, and infected teachers pass it on to their students.

The first MV was discovered by David Hestenes, a theoretical physicist, best known as chief architect of geometric algebra as a unified language for mathematics and physics.

References

David Hestenes, Mathematical Viruses